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A Computational Analysis of the Core of a Trading Economy with Three Competitive Equilibria and a Finite Number of Traders

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Abstract

In this paper we examine the structure of the core of a trading economy with three competitive equilibria as the number of traders (N) is varied. We also examine the sensitivity of the multiplicity of equilibria and of the core to variations in individual initial endowments. Computational results show that the core first splits into two pieces at N = 5 and then splits a second time into three pieces at N = 12. Both of these splits occur not at a point but as a contiguous gap. As N is increased further, the core shrinks by N = 600 with essentially only the 3 competitive equilibria remaining. We find that the speed of convergence of the core toward the three competitive equilibria is not uniform. Initially, for small N, it is not of the order 1/N but when N is large, the convergence rate is approximately of the order 1/N. Small variations in the initial individual endowments along the price rays to the competitive equilibria make the respective competitive equilibrium (CE) unique and once a CE becomes unique, it remains so for all allocations on the price ray. Sensitivity analysis of the core reveals that in the large part of the endowment space where the competitive equilibrium is unique, the core either converges to the single CE or it splits into two segments, one of which converges to the CE and the other disappears.

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  • Martin Shubik & Alok Kumar, 2001. "A Computational Analysis of the Core of a Trading Economy with Three Competitive Equilibria and a Finite Number of Traders," Cowles Foundation Discussion Papers 1290, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1290
    Note: CFP 1135.
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    References listed on IDEAS

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    1. Shapley, Lloyd S & Shubik, Martin, 1977. "An Example of a Trading Economy with Three Competitive Equilibria," Journal of Political Economy, University of Chicago Press, vol. 85(4), pages 873-875, August.
    2. Shapley, L S, 1975. "An Example of a Slow-Converging Core," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 16(2), pages 345-351, June.
    3. Shapley, Lloyd S & Shubik, Martin, 1969. "Pure Competition, Coalitional Power, and Fair Division," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 10(3), pages 337-362, October.
    4. Gerard Debreu, 1963. "On a Theorem of Scarf," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 30(3), pages 177-180.
    5. Debreu, Gerard, 1975. "The rate of convergence of the core of an economy," Journal of Mathematical Economics, Elsevier, vol. 2(1), pages 1-7, March.
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    More about this item

    Keywords

    Core; Multiple competitive equilibria; Speed of convergence; Sensitivity Analysis;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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